According to a report in The New York Times (Warning: source may be paywalled), Ajit Pai and the FCC approved a set of rules in 2017 to allow television broadcasters to increase the number of stations they own. Weeks after the rules were approved, Sinclair Broadcasting announced a $3.9 billion deal to buy Tribune Media. PC Gamer reports: The deal was made possible by the new set of rules, which subsequently raised some eyebrows. Notably, the FCC's inspector general is reportedly investigating if Pai and his aides abused their position by pushing for the rule changes that would make the deal possible, and timing them to benefit Sinclair. The extent of the investigation is not clear, nor is how long it will take. However, it does bring up the question of whether Pai had coordinated with Sinclair, and it could force him to publicly address the topic, which he hasn't really done up to this point.
Legislators first pushed for an investigation into this matter last November. At the time, a s

The shallow water equations (SWE) are a widely used model for the propagation
of surface waves on the oceans. In particular, the SWE are used to model the
propagation of tsunami waves in the open ocean. We consider the associated data
assimilation problem of optimally determining the initial conditions for the
one-dimensional SWE in an unbounded domain from a small set of observations of
the sea surface height and focus on how the structure of the observation
operator affects the convergence of the gradient approach employed to solve the
data assimilation problem computationally. In the linear case we prove a
theorem that gives sufficient conditions for convergence to the true initial
conditions. It asserts that at least two observation points must be used and at
least one pair of observation points must be spaced more closely than half the
effective minimum wavelength of the energy spectrum of the initial conditions.
Our analysis is confirmed by numerical experiments for both the line

A growing number of Coinbase customers are complaining that the cryptocurrency exchange withdrew unauthorized money out of their accounts. From a report: In some cases, this drained their linked bank accounts below zero, resulting in overdraft charges. In a typical anecdote posted on Reddit, one user said they purchased Bitcoin, Ether, and Litecoin for a total of $300 on February 9th. A few days later, the transactions repeated five times for a total of $1,500, even though the user had not made any more purchases. That was enough to clear out this user's bank account, they said, resulting in fees. [...] Coinbase representatives have been responding to similar complaints on Reddit for about two weeks, but the volume of complaints seems to have spiked over the last 24 hours. Similar complaints have popped up on forums and Twitter.

TorrentFreak: As entertainment companies and Internet services spar over the boundaries of copyright law, the EFF is urging the US Copyright Office to keep "copyright's safe harbors safe." In a petition just filed with the office, the EFF warns that innovation will be stymied if Congress goes ahead with a plan to introduce proactive 'piracy' filters at the expense of the DMCA's current safe harbor provisions. [...] "Major media and entertainment companies and their surrogates want Congress to replace today's DMCA with a new law that would require websites and Internet services to use automated filtering to enforce copyrights. "Systems like these, no matter how sophisticated, cannot accurately determine the copyright status of a work, nor whether a use is licensed, a fair use, or otherwise non-infringing. Simply put, automated filters censor lawful and important speech," the EFF warns.

Linux Journal takes a look at the newly announced LinuxBoot project. LWN covered a related talk back in November. "Modern firmware generally consists of two main parts: hardware initialization (early stages) and OS loading (late stages). These parts may be divided further depending on the implementation, but the overall flow is similar across boot firmware. The late stages have gained many capabilities over the years and often have an environment with drivers, utilities, a shell, a graphical menu (sometimes with 3D animations) and much more. Runtime components may remain resident and active after firmware exits. Firmware, which used to fit in an 8 KiB ROM, now contains an OS used to boot another OS and doesn't always stop running after the OS boots. LinuxBoot replaces the late stages with a Linux kernel and initramfs, which are used to load and execute the next stage, whatever it may be and wherever it may come from. The Linux kernel included in LinuxBoot is called the 'boot ke

An anonymous reader quotes a report from TechCrunch: The ARCEP, France's equivalent of the FCC in the U.S., wants to go beyond telecommunications companies. While many regulatory authorities have focused on carriers and internet service providers, the French authority thinks Google, Apple, Amazon and all the big tech companies also need their own version of net neutrality. The ARCEP just published a thorough 65-page report about the devices we use every day. The report says that devices give you a portion of the internet and prevent an open internet. "With net neutrality, we spend all our time cleaning pipes, but nobody is looking at faucets," ARCEP president Sebastien Soriano told me. "Everybody assumes that the devices that we use to go online don't have a bias. But if you want to go online, you need a device just like you need a telecom company."
Now that net neutrality has been laid down in European regulation, the ARCEP has been looking at devices for the past couple of years. And

We solve the simultaneous conjugacy problem in Artin's braid groups and, more
generally, in Garside groups, by means of a complete, effectively computable,
finite invariant. This invariant generalizes the one-dimensional notion of
super summit set to arbitrary dimensions. One key ingredient in our solution is
the introduction of a provable high-dimensional version of the Birman--Ko--Lee
cycling theorem. The complexity of this solution is a small degree polynomial
in the cardinalities of our generalized super summit sets and the input
parameters. Computer experiments suggest that the cardinality of this
invariant, for a list of order $N$ independent elements of Artin's braid group
$B_N$, is generically close to~1.

We show that allowing magnetic fields to be complex-valued leads to an
improvement in the magnetic Hardy-type inequality due to Laptev and Weidl. The
proof is based on the study of momenta on the circle with complex magnetic
fields, which is of independent interest in the context of PT-symmetric and
quasi-Hermitian quantum mechanics. We study basis properties of the
non-self-adjoint momenta and derive closed formulae for the similarity
transforms relating them to self-adjoint operators.

This paper considers how to obtain MCMC quantitative convergence bounds which
can be translated into tight complexity bounds in high-dimensional setting. We
propose a modified drift-and-minorization approach, which establishes a
generalized drift condition defined in a subset of the state space. The subset
is called the "large set", and is chosen to rule out some "bad" states which
have poor drift property when the dimension gets large. Using the "large set"
together with a "centered" drift function, a quantitative bound can be obtained
which can be translated into a tight complexity bound. As a demonstration, we
analyze a certain realistic Gibbs sampler algorithm and obtain a complexity
upper bound for the mixing time, which shows that the number of iterations
required for the Gibbs sampler to converge is constant. It is our hope that
this modified drift-and-minorization approach can be employed in many other
specific examples to obtain complexity bounds for high-dimensional Markov
ch

This thesis consists of two parts.
Part I is an introduction to Hermite processes, Hermite random fields, Fisher
information and to the papers constituting the thesis. More precisely, in
Section 1 we introduce Hermite processes in a nutshell, as well as some of its
basic properties. It is the necessary background for the articles [a] and [c].
In Section 2 we consider briefly the multiparameter Hermite random fields and
we study some less elementary facts which are used in the article [b]. In
section 3, we recall some terminology about Fisher information related to the
article [d]. Finally, our articles [a] to [d] are summarised in Section 4.
Part II consists of the articles themselves:
[a] T.T. Diu Tran (2017): Non-central limit theorem for quadratic functionals
of Hermite-driven long memory moving average processes. \textit{Stochastic and
Dynamics}, \textbf{18}, no. 4.
[b] T.T. Diu Tran (2016): Asymptotic behavior for quadratic variations of
non-Gaussian multiparameter Hermite random

We consider convex hypersurfaces with boundary which meet a strictly convex
cone perpendicularly. If those hypersurfaces expand inside this cone by the
power of the Gauss curvature, we prove that this evolution exists for all the
time and the evolving hypersurfaces converge smoothly to a piece of round
sphere after rescaling.

It is established that if a harmonic function $u$ on the unit disk $\mathbb
D$ in $\mathbb C$ has angular limits on a measurable set $E$ of the unit circle
$\partial\mathbb D$, then its conjugate harmonic function $v$ in $\mathbb D$
also has angular limits a.e. on $E$ and both boundary functions are finite a.e.
and measurable on $E$. The result is extended to arbitrary Jordan domains with
rectifiable boundaries in terms of angular limits and of the natural parameter.

We show how to build an immersion coupling of a two-dimensional Brownian
motion $(W_1, W_2)$ along with $\binom{n}{2} + n= \tfrac12n(n+1)$ integrals of
the form $\int W_1^iW_2^j \circ dW_2$, where
$j=1,\ldots,n$ and $i=0, \ldots, n-j$ for some fixed $n$. The resulting
construction is applied to the study of couplings of certain hypoelliptic
diffusions (driven by two-dimensional Brownian motion using polynomial vector
fields). This work follows up previous studies concerning coupling of Brownian
stochastic areas and time integrals (Ben Arous, Cranston and Kendall (1995),
Kendall and Price (2004), Kendall (2007), Kendall (2009), Kendall (2013),
Banerjee and Kendall (2015), Banerjee, Gordina and Mariano (2016)), and is part
of an ongoing research programme aimed at gaining a better understanding of
when it is possible to couple not only diffusions but also multiple selected
integral functionals of the diffusions.

This paper is about the family of smooth quartic surfaces $X \subset
\mathbb{P}^3$ that are invariant under the Heisenberg group $H_{2,2}$. For a
very general such surface $X$, we show that the Picard number of $X$ is 16 and
determine its Picard group. It turns out that the general Heisenberg invariant
quartic contains 320 smooth conics and that in the very general case, this
collection of conics generates the Picard group.

We prove a number of statistical properties of Hecke coefficients for unitary
cuspidal representations on $\operatorname{GL}(2)$ over number fields
(unconditionally) and on $\operatorname{GL}(n)$ over number fields
(conditionally, either assuming the Ramanujan conjecture, or the functoriality
of $\pi\otimes\pi^\vee$). Using partial bounds on Hecke coefficients,
properties of Rankin-Selberg $L$-functions, and instances of Langlands
functoriality, we obtain bounds on the set of places where (linear combinations
of) Hecke coefficients are bounded above (or below). We furthermore prove a
number of consequences: we obtain an improved answer to a question of Serre
about the occurrence of large Hecke eigenvalues of Maass forms ($|a_p|>1$ for
density at least $0.00135$ set of primes), we prove the existence of negative
Hecke coefficients over arbitrary number fields, and we obtain distributional
results on the Hecke coefficients $a_v$ when $v$ varies in certain congruence
or Galois classes.

According to Mashable, Facebook account holder Gabriel Lewis tweeted that Facebook texted "spam" to the phone number he submitted for the purposes of 2-factor authentication. Lewis insists that he did not have mobile notifications turned on, and when he replied "stop" and "DO NOT TEXT ME," he says those messages showed up on his Facebook wall. From the report: Lewis explained his version of the story to Mashable via Twitter direct message. "[Recently] I decided to sign up for 2FA on all of my accounts including FaceBook, shortly afterwards they started sending me notifications from the same phone number. I never signed up for it and I don't even have the FB app on my phone." Lewis further explained that he can go "for months" without signing into Facebook, which suggests the possibility that Mark Zuckerberg's creation was feeling a little neglected and trying to get him back. According to Lewis, he signed up for 2FA on Dec. 17 and the alleged spamming began on Jan. 5. Importantly, Lewi

The heads of six top U.S. intelligence agencies told the Senate Intelligence Committee on Tuesday they would not advise Americans to use products or services from Chinese smartphone maker Huawei. "The six -- including the heads of the CIA, FBI, NSA and the director of national intelligence -- first expressed their distrust of Apple-rival Huawei and fellow Chinese telecom company ZTE in reference to public servants and state agencies," reports CNBC. From the report: "We're deeply concerned about the risks of allowing any company or entity that is beholden to foreign governments that don't share our values to gain positions of power inside our telecommunications networks," FBI Director Chris Wray testified. "That provides the capacity to exert pressure or control over our telecommunications infrastructure," Wray said. "It provides the capacity to maliciously modify or steal information. And it provides the capacity to conduct undetected espionage."
In a response, Huawei said that it "pos

Some new directions to lay a rigorous mathematical foundation for the
phase-portrait-based modelling of fingerprints are discussed in the present
work. Couched in the language of dynamical systems, and preparing to a
preliminary modelling, a back-to-basics analogy between Poincar\'{e}'s
categories of equilibria of planar differential systems and the basic
fingerprint singularities according to Purkyn\v{e}-Galton's standards is first
investigated. Then, the problem of the global representation of a fingerprint's
flow-like pattern as a smooth deformation of the phase portrait of a
differential system is addressed. Unlike visualisation in fluid dynamics, where
similarity between integral curves of smooth vector fields and flow streamline
patterns is eye-catching, the case of an oriented texture like a fingerprint's
stream of ridges proved to be a hard problem since, on the one hand, not all
fingerprint singularities and nearby orientational behaviour can be modelled by
canonical phase por

Let f be local diffeomorphism between real Banach spaces. We prove that if
the locally Lipschitz functional F(x)=1/2|f(x)-y|^2 satisfies the Chang
Palais-Smale condition for all y in the target space of f, then f is a
norm-coercive global diffeomorphism. We also give a version of this fact for a
weighted Chang Palais-Smale condition. Finally, we study the relationship of
this criterion to some classical global inversion conditions.

Kuramoto oscillators are widely used to explain collective phenomena in
networks of coupled oscillatory units. We show that simple networks of two
populations with a generic coupling scheme can exhibit chaotic dynamics as
conjectured by Ott and Antonsen [Chaos, 18, 037113 (2008)]. These chaotic mean
field dynamics arise universally across network size, from the continuum limit
of infinitely many oscillators down to very small networks with just two
oscillators per population. Hence, complicated dynamics are expected even in
the simplest description of oscillator networks.

We prove some Schur positivity results for the chromatic symmetric function
$X_G$ of a (hyper)graph $G$, using connections to the group algebra of the
symmetric group. The first such connection works for (hyper)forests $F$: we
describe the Schur coefficients of $X_F$ in terms of eigenvalues of a product
of Hermitian idempotents in the group algebra, one factor for each edge (a more
general formula of similar shape holds for all chordal graphs). Our main
application of this technique is to prove a conjecture of Taylor on the Schur
positivity of certain $X_F$, which implies Schur positivity of the formal group
laws associated to various combinatorial generating functions. Second, we
introduce the pointed chromatic symmetric function $X_{G,v}$ associated to a
rooted graph $(G,v)$. We prove that if $X_{G,v}$ and $X_{H,w}$ are positive in
the generalized Schur basis of Strahov, then the chromatic symmetric function
of the wedge sum of $(G,v)$ and $(H,w)$ is Schur positive.

That's according to the Y Combinator-backed real-estate startup Open Listings, which looked at median home sales prices near the headquarters (meaning within a 20-minute commute) of some of the Bay Area's biggest and best-known tech companies. Fast Company: Using public salary data from Paysa, Open Listings then looked at how many software engineers from those companies could actually afford to buy a house close to their office. Here's what it found: Engineers at five major SF-based tech companies would need to spend over the 28% threshold of their income to afford a monthly mortgage near their offices. Apple engineers would have to pay an average of 33% of their monthly income for a mortgage near work. That's the highest percentage of the companies analyzed, and home prices in Cupertino continue to skyrocket. Google wasn't much better at 32%, and living near the Facebook office would cost an engineer 29% of their monthly paycheck.

We examine the probability that at least two eigenvalues of an Hermitian
matrix-valued Gaussian process, collide. In particular, we determine sharp
conditions under which such probability is zero. As an application, we show
that the eigenvalues of a real symmetric matrix-valued fractional Brownian
motion of Hurst parameter $H$, collide when $H<1/2$ and don't collide when
$H>\frac{1}{2}$, while those of a complex Hermitian fractional Brownian motion
collide when $H<\frac{1}{3}$ and don't collide when $H>\frac{1}{3}$. Our
approach is based on the relation between hitting probabilities for Gaussian
processes with the capacity and Hausdorff dimension of measurable sets.

In this paper our aim is to find the radii of starlikeness and convexity of
the nth derivative of Bessel function for three different kind of
normalization. The key tools in the proof of our main results are the
Mittag-Leffler expansion for nth derivative of Bessel function and properties
of real zeros of it. The main results of the paper are natural extensions of
some known results on classical Bessel functions of the first kind.

A general fractional relaxation equation is considered with a convolutional
derivative in time introduced by A. Kochubei (Integr. Equ. Oper. Theory 71
(2011), 583-600). This equation generalizes the single-term, multi-term and
distributed-order fractional relaxation equations. The fundamental and the
impulse-response solutions are studied in detail. Properties such as
analyticity and subordination identities are established and employed in the
proof of an upper and a lower bound. The obtained results extend some known
properties of the Mittag-Leffler functions. As an application of the estimates,
uniqueness and conditional stability are established for an inverse source
problem for the general time-fractional diffusion equation on a bounded domain.

Many physical problems can be cast in a form where a constitutive equation
${\bf J(x)}={\bf L(x)E(x)}+{\bf h(x)}$ with a source term ${\bf h(x)}$ holds
for all ${\bf x}\in R^d$ and relates fields ${\bf E}$ and ${\bf J}$ that
satisfy appropriate differential constraints, symbolized by ${\bf E\in\cal E}$
and ${\bf J}\in\cal J$ where $\cal E$ and $\cal J$ are orthogonal spaces that
span the space $\cal H$ of square-integrable fields in which ${\bf h}$ lies.
Here we show that if the moduli ${\bf L(x)}$ are constrained to take values in
certain nonlinear manifolds $\cal M$, and satisfy suitable coercivity and
boundedness conditions, then the infinite body Green's function for the problem
satisfies certain exact identities. A corollary of our theory is that it also
provides the framework for establishing links between the Green's functions for
different physical problems, sharing some commonality in their geometry. The
analysis is based on the theory of exact relations for composites, but, u

Let $(M,g)$ be a compact Riemann surfaces, and $\pi:P\to M$ be a principal
$\mathbb S^1$-bundle over $M$ endowed with a connection $A$. Fixing an inner
product on the Lie algebra of $\mathbb{S}^1$, the connection $A$ and metric $g$
define a Riemannian metric $g_A$ on $P$. In this article, we show that the
Ricci flow equation of metric $g_A$ is equivalent to a system of differential
equations. We will give an explicit solution of normalized Ricci flow equation
of metric $g_A$ in the case where the base manifold is of constant curvature
and the the initial connection $A$ is Yang-Mills. Finally, we will describe
some asymptotic behaviors of these flows.

It is important in many applications to be able to extend the (outer) unit
normal vector field from a hypersurface to its neighborhood in such a way that
the result is a unit gradient field. The aim of the paper is to provide an
elementary proof of the existence and uniqueness of such an extension.

A problem of further generalization of generalized Choi maps $\Phi_{[a,b,c]}$
acting on $\mathbb{M}_3$ introduced by Cho, Kye and Lee is discussed. Some
necessary conditions for positivity of the generalized maps are provided as
well as some sufficient conditions. Also some sufficient condition for
decomposability of these maps is shown.

In this article, we derive the existence of positive solutions of a
semi-linear, non-local elliptic PDE, involving a singular perturbation of the
fractional laplacian, coming from the fractional Hardy-Sobolev-Maz'ya
inequality, derived in this paper. We also derive symmetry properties and a
precise asymptotic behaviour of solutions.

We consider the generalized Egorov's statement (Egorov's Theorem without the
assumption on measurability of the functions, see \cite{tw:nget}) in the case
of an ideal convergence and a number of different types of ideal convergence
notion. We prove that in those cases the generalized Egorov's statement is
independent from ZFC.

We generalize Banaszczyk's seminal tail bound for the Gaussian mass of a
lattice to a wide class of test functions. We therefore obtain quite general
transference bounds, as well as bounds on the number of lattice points
contained in certain bodies. As example applications, we bound the lattice
kissing number in $\ell_p$ norms by $e^{(n+ o(n))/p}$ for $0 < p \leq 2$, and
also give a proof of a new transference bound in the $\ell_1$ norm.

Given a prime $p$ and an integer $d>1$, we give a numerical criterion to
decide whether the $\ell$-adic sheaf associated to the one-parameter
exponential sums $t\mapsto \sum_x\psi(x^d+tx)$ over ${\mathbb F}_p$ has finite
monodromy or not, and work out some explicit cases where this is computable.

In the present paper, the structure of a finite group $G$ having a nonnormal
T.I. subgroup $H$ which is also a Hall $\pi$-subgroup is studied. As a
generalization of a result due to Gow, we prove that $H$ is a Frobenius
complement whenever $G$ is $\pi$-separable. This is achieved by obtaining the
fact that Hall T.I. subgroups are conjugate in a finite group. We also prove
two theorems about normal complements one of which generalizes a classical
result of Frobenius.

Motivated by matrix recovery problems such as Robust Principal Component
Analysis, we consider a general optimization problem of minimizing a smooth and
strongly convex loss applied to the sum of two blocks of variables, where each
block of variables is constrained or regularized individually. We present a
novel Generalized Conditional Gradient method which is able to leverage the
special structure of the problem to obtain faster convergence rates than those
attainable via standard methods, under a variety of interesting assumptions. In
particular, our method is appealing for matrix problems in which one of the
blocks corresponds to a low-rank matrix and avoiding prohibitive full-rank
singular value decompositions, which are required by most standard methods, is
most desirable. Importantly, while our initial motivation comes from problems
which originated in statistics, our analysis does not impose any statistical
assumptions on the data.

Let $M$ be complex projective manifold, and $A$ a positive line bundle on it.
Assume that a compact and connected Lie group $G$ acts on $M$ in a Hamiltonian
manner, and that this action linearizes to $A$. Then there is an associated
unitary representation of $G$ on the associated algebro-geometric Hardy space.
If the moment map is nowhere vanishing, the isotypical component are all finite
dimensional; they are generally not spaces of sections of some power of $A$.
One is then led to study the local and global asymptotic properties the
isotypical component associated to a weight $k \, \boldsymbol{ \nu }$, when
$k\rightarrow +\infty$. In this paper, part of a series dedicated to this
general theme, we consider the case $G=U(2)$.

Many smoothness spaces in harmonic analysis are decomposition spaces. In this
paper we ask: Given two decomposition spaces, is there an embedding between the
two?
A decomposition space $\mathcal{D}(\mathcal{Q}, L^p, Y)$ can be described
using : a covering $\mathcal{Q}=(Q_{i})_{i\in I}$ of the frequency domain, an
exponent $p$ and a sequence space $Y\subset\mathbb{C}^{I}$. Given these, the
decomp. space norm of a distribution $g$ is $\| g\| _{\mathcal{D}(\mathcal{Q},
L^p, Y)}=\left\| \left(\left\|
\mathcal{F}^{-1}\left(\varphi_{i}\widehat{g}\right)\right\|
_{L^{p}}\right)_{i\in I}\right\| _{Y}$, where $(\varphi_{i})_{i\in I}$ is a
suitable partition of unity for $\mathcal{Q}$.
We establish readily verifiable criteria which ensure an embedding
$\mathcal{D}(\mathcal{Q}, L^{p_1}, Y)\hookrightarrow\mathcal{D}(\mathcal{P},
L^{p_2}, Z)$, mostly concentrating on the case, $Y=\ell_{w}^{q_{1}}(I)$ and
$Z=\ell_{v}^{q_{2}}(J)$.
The relevant sufficient conditions are $p_{1}\leq p_{2}$, and finitene

X-ray Computed Tomography (CT) reconstruction from a sparse number of views
is a useful way to reduce either the radiation dose or the acquisition time,
for example in fixed-gantry CT systems, however this results in an ill-posed
inverse problem whose solution is typically computationally demanding.
Approximate Message Passing (AMP) techniques represent the state of the art for
solving undersampling Compressed Sensing problems with random linear
measurements but there are still not clear solutions on how AMP should be
modified and how it performs with real world problems. This paper investigates
the question of whether we can employ an AMP framework for real sparse view CT
imaging? The proposed algorithm for approximate inference in tomographic
reconstruction incorporates a number of advances from within the AMP community,
resulting in the Denoising CT Generalised Approximate Message Passing algorithm
(DCT-GAMP). Specifically, this exploits the use of state of the art image
denoisers t

In Cultural Heritage (CH) imaging, data acquired within different spectral
regions are often used to inspect surface and sub-surface features. Due to the
experimental setup, these images may suffer from intensity inhomogeneities,
which may prevent conservators from distinguishing the physical properties of
the object under restoration. Furthermore, in multi-modal imaging, the transfer
of information between one modality to another is often used to integrate image
contents. In this paper, we apply the image osmosis model proposed in (Weickert
et al. 2013) to solve similar problems arising when using diagnostic CH imaging
techniques based on reflectance, emission and fluorescence mode in the optical
and thermal range. For an efficient computation, we use stable operator
splitting techniques. We test our methods on real artwork datasets: the thermal
measurements of the mural painting "Monocromo" by Leonardo Da Vinci, the
UV-VIS-IR imaging of an ancient Russian icon and the Archimedes Pali

Starting with a Nambu-Goto action, a Dirac-like equation can be constructed
by taking the square-root of the momentum constraint. The eigenvalues of the
resulting Hamiltonian are real and correspond to masses of the excited string.
In particular there are no tachyons. A special case of radial oscillations of a
closed string in Minkowski space-time admits exact solutions in terms of wave
functions of the harmonic oscillator.

We consider the quotient variety associated to a linear representation of the
cyclic group of order p in characteristic p>0. We estimate the minimal
discrepancy of exceptional divisors over the singular locus. In particular, we
give criteria for the quotient variety being terminal, canonical and log
canonical. As an application, we obtain new examples of non-Cohen-Macaulay
terminal singularities, adding to examples recently announced by Totaro.

In this paper we study continuous parametrized families of dissipative flows,
which are those flows having a global attractor. The main motivation for this
study comes from the observation that, in general, global attractors are not
robust, in the sense that small perturbations of the flow can destroy their
globality. We give a necessary and sufficient condition for a global attractor
to be continued to a global attractor. We also study, using shape theoretical
methods and the Conley index, the bifurcation global to non-global.

Design of energy efficient protocols for modern wireless systems has become
an important area of research. In this paper, we propose a distributed
optimization algorithm for the channel assignment problem for multiple
interfering transceiver pairs that cannot communicate with each other. We first
modify the auction algorithm for maximal energy efficiency and show that the
problem can be solved without explicit message passing using the carrier sense
multiple access (CSMA) protocols. We then develop a novel scheme by converting
the channel assignment problem into perfect matchings on bipartite graphs. The
proposed scheme improves the energy efficiency and does not require any
explicit message passing or a shared memory between the users. We derive bounds
on the convergence rate and show that the proposed algorithm converges faster
than the distributed auction algorithm and achieves near-optimal performance
under Rayleigh fading channels. We also present an asymptotic performance
analysi

This paper is concerned with efficiently coloring sparse graphs in the
distributed setting with as few colors as possible. According to the celebrated
Four Color Theorem, planar graphs can be colored with at most 4 colors, and the
proof gives a (sequential) quadratic algorithm finding such a coloring. A
natural problem is to improve this complexity in the distributed setting. Using
the fact that planar graphs contain linearly many vertices of degree at most 6,
Goldberg, Plotkin, and Shannon obtained a deterministic distributed algorithm
coloring $n$-vertex planar graphs with 7 colors in $O(\log n)$ rounds. Here, we
show how to color planar graphs with 6 colors in $\mbox{polylog}(n)$ rounds.
Our algorithm indeed works more generally in the list-coloring setting and for
sparse graphs (for such graphs we improve by at least one the number of colors
resulting from an efficient algorithm of Barenboim and Elkin, at the expense of
a slightly worst complexity). Our bounds on the number of colo

The Gram spectrahedron $\text{Gram}(f)$ of a form $f$ with real coefficients
parametrizes the sum of squares decompositions of $f$, modulo orthogonal
equivalence. For $f$ a sufficiently general positive binary form of arbitrary
degree, we show that $\text{Gram}(f)$ has extreme points of all ranks in the
Pataki range. This is the first example of a family of spectrahedra of
arbitrarily large dimensions with this property. We also calculate the
dimension of the set of rank $r$ extreme points, for any $r$. Moreover, we
determine the pairs of rank two extreme points for which the connecting line
segment is an edge of $\text{Gram}(f)$.

We apply methods of proof mining to obtain uniform quantitative bounds on the
strong convergence of the proximal point algorithm for finding minimizers of
convex, lower semicontinuous proper functions in CAT(0) spaces. Thus, for
uniformly convex functions we compute rates of convergence, while, for totally
bounded CAT(0) spaces we apply methods introduced by Kohlenbach, the first
author and Nicolae to compute rates of metastability.

In Artin-Tits groups attached to Coxeter groups of spherical type, we give a
combinatorial formula to express the simple elements of the dual braid monoids
in the classical Artin generators. Every simple dual braid is obtained by
lifting an $S$-reduced expression of its image in the Coxeter group, in a way
which involves Reading's $c$-sortable elements. It has as an immediate
consequence that simple dual braids are Mikado braids (the known proofs of this
result either require topological realizations of the Artin groups or
categorification techniques), and hence that their images in the Iwahori-Hecke
algebras have positivity properties. In the classical types, this requires to
give an explicit description of the inverse of Reading's bijection from
$c$-sortable elements to noncrossing partitions of a Coxeter element $c$, which
might be of independent interest. The bijections are described in terms of the
noncrossing partition models in these types. While the proof of the formula is
case

We propose the Roe C*-algebra from coarse geometry as a model for topological
phases of disordered materials. We explain the robustness of this C*-algebra
and formulate the bulk-edge correspondence in this framework. We describe the
map from the K-theory of the group C*-algebra of Z^d to the K-theory of the Roe
C*-algebra, both for real and complex K-theory.

In this letter, we study the economic dispatch with adjustable transformer
ratio and phase shifter, both of which, along with the transmission line, are
formulated into a generalized branch model. Resulted nonlinear parts are
thereafter exactly linearized using the piecewise liner technique to make the
derived ED problem computationally tractable. Numerical studies based on
modified IEEE systems demonstrate the effectiveness of the proposed method to
efficiency and flexibility of power system operation.

In this paper we prove that all small, uniformly in time $L^1\cap H^1$
bounded solutions to KdV and related perturbations must converge to zero, as
time goes to infinity, locally in an increasing-in-time region of space of
order $t^{1/2}$ around any compact set in space. This set is included in the
linearly dominated dispersive region $x\ll t$. Moreover, we prove this result
independently of the well-known supercritical character of KdV scattering. In
particular, no standing breather-like nor solitary wave structures exists in
this particular regime. For the proof, we make use of well-chosen weighted
virial identities. The main new idea employed here with respect to previous
results is the fact that the $L^1$ integral is subcritical with respect to the
KdV scaling.

A cluster of $n$ needles ($1\leq n<\infty$) is dropped at random onto a plane
lattice of rectangles. Each needle is fixed at one end in the cluster centre
and can rotate independently about this centre. The distribution of the
relative number of needles intersecting the lattice is shown to converge
uniformly to the limit distribution as $n\rightarrow\infty$.

We show any matrix of rank $r$ over $\mathbb{F}_q$ can have $\leq
\binom{r}{k}(q-1)^k$ distinct columns of weight $k$ if $ k \leq O(\sqrt{\log
r})$ (up to divisibility issues), and $\leq \binom{r}{k}(q-1)^{r-k}$ distinct
columns of co-weight $k$ if $k \leq O_q(r^{2/3})$. This shows the natural
examples consisting of only $r$ rows are optimal for both, and the proofs will
recover some form of uniqueness of these examples in all cases.

The article is devoted to explicit one-step numerical methods with strong
order of convergence 2.5 for Ito stochastic differential equations with
multidimensional non-additive noise. We consider the numerical methods, based
on the unified Taylor-Ito and Taylor-Stratonovich expansions. For numerical
modeling of multiple Ito and Stratonovich stochastic integrals of
multiplicities 1-5 we appling the method of multiple Fourier-Legendre series,
converging in the mean-square sense in the space $L_2([t, T]^k);$
$k=1,\ldots,5$. The article is addressed to engineers who use numerical
modeling in stochastic control and for solving the non-linear filtering
problem. The article can be interesting for the scientists who working in the
field of numerical integration of stochastic differential equations.

Frechet bounds of the 1-st kind for sets of events and its main properties
are considered. The lemma on not more than two nonzero values of lower
Frechet-bounds of the 1-st kind for a set of half-rare events is proved with
the corollary on the analogous assertion for sets of events with arbitrary
event-probability distributions.

A crucial limitation of current high-resolution 3D photoacoustic tomography
(PAT) devices that employ sequential scanning is their long acquisition time.
In previous work, we demonstrated how to use compressed sensing techniques to
improve upon this: images with good spatial resolution and contrast can be
obtained from suitably sub-sampled PAT data acquired by novel acoustic scanning
systems if sparsity-constrained image reconstruction techniques such as total
variation regularization are used. Now, we show how a further increase of image
quality or acquisition speed rate can be achieved for imaging dynamic processes
in living tissue (4D PAT). The key idea is to exploit the additional temporal
redundancy of the data by coupling the previously used spatial image
reconstruction models with sparsity-constrained motion estimation models. While
simulated data from a two-dimensional numerical phantom will be used to
illustrate the main properties of this recently developed
joint-image-recons

Submodular functions have applications throughout machine learning, but in
many settings, we do not have direct access to the underlying function $f$. We
focus on stochastic functions that are given as an expectation of functions
over a distribution $P$. In practice, we often have only a limited set of
samples $f_i$ from $P$. The standard approach indirectly optimizes $f$ by
maximizing the sum of $f_i$. However, this ignores generalization to the true
(unknown) distribution. In this paper, we achieve better performance on the
actual underlying function $f$ by directly optimizing a combination of bias and
variance. Algorithmically, we accomplish this by showing how to carry out
distributionally robust optimization (DRO) for submodular functions, providing
efficient algorithms backed by theoretical guarantees which leverage several
novel contributions to the general theory of DRO. We also show compelling
empirical evidence that DRO improves generalization to the unknown stochastic
submod

Reliable and efficient spectrum sensing through dynamic selection of a subset
of spectrum sensors is studied. The problem of selecting K sensor measurements
from a set of M potential sensors is considered where K << M. In addition, K
may be less than the dimension of the unknown variables of estimation. Through
sensor selection, we reduce the problem to an under-determined system of
equations with potentially infinite number of solutions. However, the sparsity
of the underlying data facilitates limiting the set of solutions to a unique
solution. Sparsity enables employing the emerging compressive sensing
technique, where the compressed measurements are selected from a large number
of potential sensors. This paper suggests selecting sensors in a way that the
reduced system of equations constructs a well-conditioned measurement matrix.
Our criterion for sensor selection is based on E-optimalily, which is highly
related to the restricted isometry property that provides some guarante